Background

The U.S. Fish and Wildlife Service (FWS) uses a Bayesian framework to predict eagle collisions and fatalities at proposed wind energy sites. One of the appeals of Bayesian modeling is the ability to incorporate prior information and uncertainty regarding parameter values into estimates, ideally providing a better representation of reality. However, there is still debate over the appropriate choice and use of priors. Additionally, as Bayesian methodology is still gaining widespread familiarity, the ways in which prior information affects model outcomes can be unclear.

The purpose of this paper is to illustrate and quantify the effects of Bayesian priors used by FWS on the predicted outcomes of the eagle mortality model, in terms of predicted eagle fatalities. We use both empirical data from wind farms provided in a study presenting the methods used by FWS to implement the Bayesian prediction model (Bay et al. 2016), as well as a simulated dataset to demonstrate these effects both in theory and in practice.

Fully understanding the effects of priors is relevant to both energy developers and wildlife conservationists, because estimates of eagle take are used to determine permitting and mitigation costs.

Bayesian Model

The components of the Bayesian model used by FWS to predict eagle fatality are:

  • Eagle Exposure Rate: minutes per hour that eagles are estimated to fly within risk areas around turbines where collisions may occur.
  • Collision Rate: probability an eagle flying within a turbine risk area is struck and killed.
  • ExpansionFactor: constant converting fatality rate to annual fatalities, based on the size of a wind facility.

These parameters are represented as probability distributions, and combined to produce a posterior distribution representing the relative likelihood of how many ealges will be killed annually at a site.

\(Fatalities = Exposure * Collision Rate * Expansion\)

Figure 1. Demonstration output of Bayesian eagle fatality model.

To be conservative, FWS uses the 80th percentile of this posterior distribution in setting take limits, and we display these values unless otherwise noted.

In this paper, we focus on eagle exposure rate, as this parameter is subject to both site-specific survey data, and a Bayesian prior distribution. The expansion factor is determined completely by the characteristics of a site. Similarly, the collision rate is estimated entirely by a prior distribution based on rates observed at existing wind facilities.

Survey Data

The data shown in Table 1 are the key values used as inputs for the eagle wind mortality model. These were derived from an empirical dataset (Appendix A in Bay et al. (2016)), which contains pre-construction eagle survey data from 26 wind energy sites. Survey effort indicates the amount of time and area covered during pre-construction surveys, and eagle observations are the duration of time over which eagles were observed flying within survey areas.

Table 1. Data from Bay et al. (2016). Effort was calculated as the product of survey area (km 2 ) and observation time (hr), derived from Appendix A.



Combining Prior Information with Survey Data

Eagle flight time and survey effort are used to estimate eagle exposure at a given wind site. Figure 2 shows how the measured exposure (green lines) is integrated with a consistent prior distribution of exposure probabilities (purple), to produce a probability distribution of exposure (yellow). This resulting distribution is used as the exposure rate to predict eagle fatalities at a site. The effect of the prior distribution is to modify extreme observed values that differ greatly from mean observations across a larger sample of sites.

Figure 2. Eagle exposure rates used to estimate predicted fatalities across from a sample of wind farms. A consistent prior probability distribution (purple) is combined with site-specific survey data (green line) to estimate a distribution of exposure probabilities for a proposed project (yellow).

Using Priors Changes Estimates

At the majority of sites, the Bayesian estimate of eagle fatalities using prior exposure information was higher than a site-specific estimate (Fig. 2). These instances are indicated by points above the 1:1 line in Figure 2. The influence of eagle exposure priors on estimated eagle fatalities was less at facilities that expended greater effort on pre-construction surveys. This pattern is indicated by the sites with the most intensive surveys falling closest to a 1:1 relationship between the two estimates (Fig. 2).

Figure 3. Relationships between estimated eagle fatalities produced incorporating prior information about eagle exposure at wind facilities, and those produced using only site-specific survey data. Plotted values are the 80th percentiles of the posterior distributions. Marker size is proportional to the size of the 80% CI produced from the posterior distribution using exposure priors.

Source: Bay et al. (2016)

Simulation Study

To investigate the effects of prior information and survey effort on predicted take, we created a dataset representing a range of possible pre-construction survey values for survey area, time, and eagle observations.

Prediction Discrepancies Increase at Extreme Values

For eagle exposure rates < 1.1 min/hr*km3, predicted fatlity rates estimated using priors were higher than site-specific estimates (i.e. above the 1:1 line in Fig. 3). This pattern reversed when exposure was > 1.1. The threshold of 1.1 corresponds to the mean exposure rate of the prior exposure distribution. The actual value of this threshold is subject to change as prior distributions are updated with new survey data, but the pattern will be consistent. Figure 4 also illustrates that predicted fatalities estimated with and without priors are more similar when survey effort is greater (i.e., larger circles).

Figure 4. Relationships between Bayesian and site-specific estimates of eagle fatalities, generated from simulated data. Plotted values are the 80th percentile of posterior fatality distributions, multiplied by the mean site scaling factor from Bay et al. 2016.

The prior distribution of exposure probabilities has a greater effect on estimates when observed exposure rate at a site is farther from the mean of the prior. Figure 5 illustrates this relationship in terms of how many standard deviations an observed exposure rate was from the prior mean . At one standard deviation above or below the mean, take estimates can differ by as much as 0.4 eagles per year.

Figure 5. Discrepancy between Bayesian and site-specific estimates of eagle fatalities, based on the difference between the observed exposure rate and prior mean (z-score). Discrepancies were calculated between the 80th percentile of the posterior fatality distributions. Values were generated from simulated data, and multiplied by the mean site scaling factor from Bay et al. 2016.

Greater Survey Effort Reduces Effect of Priors

Increased survey effort reduced the influence of prior distributions of exposure probability on estimates of fatalities, resulting in less discrepacy between predictions obtained using priors and those obtained using site survey data only. The magnitude of the effect of effort on estimate discrepancy was contingent upon the observed eagle exposure rate. This discrepancy decreased logarithmically with greater survey effort (Fig.6). Similarly, the size of 80% CI from posterior estimates decreased as survey efforts increase, indicating less uncertainty around fatality predictions.

Figure 6. Effects of survey effort on a) difference between fatalities predicted with and without prior exposure information, and b) the size of the 80% CI of the predicted fatality distribution, relative to the mean.

Methods

Empirical Data

The prior distribution on eagle exposure used by FWS is a gamma distribution, with shape and rate parameters represented by eagle flight time in minutes, and survey effort in (hr km3). Appendix A in Bay et al. (2016) initially presented survey effort data as plot area in hectares, and observation time in minutes. We converted these to Survey effort (hr km3) by multiplying plot area (ha) by 0.2 km/ha, times the reported observation time converted to hours.

We derived an expansion factor, which was not reported in Bay et al. (2016), to produce fatality estimates in numbers of eagles per year. We quantified the relationship between ‘Flight Risk Area’, and predicted ‘Collisions per Annuum’ reported in Bay et al. (2016) using a generalized linear model. First, we divided Collisions per Annuum reported at each site by the site exposure, and mean of the prior collision rate distribution to obtain the site-specific collision rates per eagle activity rate. We modeled this rate as a quadratic funciton of site risk area.

glm(data = Bay_16, (COLLISIONS/(FLIGHT_MIN/EFFORT))/0.002895415 ~ RISK_HA + RISK_HA^2)

The coefficients from this model were then used as the expansion factor for each site. Thus we calculate the flight risk area as -2.25581310^{4} + 2306.4 * Risk Area + -2.61 * (Risk Area) 2.

Simulation Data

We generated hypothetical values for survey effort based on the minimum requirements provided by U.S. FWS for pre-construction monitoring. FWS requires at least one cylindrical survey plot with radius > 800m and height > 200m, and that plots be surveyed for at least 12 hours per year, for two years. Thus, the minimum values for survey area and time were 0.402 km3, and 24 hrs, respectively. We simulated up to five plots (area = 2.01) in increments of one, and up to 240 hrs in increments of 12, and calculated survey effort for all combinations.

Observed eagle flight time (min) provided in Bay et al. (2016) is a function of survey effort and eagle activity at a site. Therefore, to generate a range of potential exposure values (min/hr*km3), we divided flight time by effort at each survey site, and took a random sample of 20 values. The final simulation data included all combinations of survey effort and eagle exposure rate, which we multiplied to obtain flight minutes, producing 1000 values.

flight <- c(0.01,0.02,0.03,0.04,0.05,
            0.06,0.07,0.08,0.09,0.10,
            0.15,0.20,0.25,0.50,0.75,
            1.00,1.50,2.00,2.50,3)
time <- seq(1, 10, 1)*12*2
area <- seq(0.402, 2.01, 0.402)
df <- expand.grid(TIME = time, AREA = area, eagle_rate = flight)
df$b <- df$TIME*df$AREA
df$a <- df$eagle_rate*df$b

Model Output

We generated posterior distributions for predicted eagle fatalities using the FWS Bayesian model. We use the updated FWS priors for collision rate, Collision ~ Beta(9.38, 3224.51), and a prior exposure probability distribution defined by the survey data, Exposure ~ Gamma(a, b), where a represents observed eagle flight minutes, and b represents survey effort, following Bay et al. (2016). We defined a and b for the exposure prior as the mean values from survey data. To obtain posterior estimates, we drew 100,000 random samples from each of these distributions and collected their products to form the posterior distribution of eagle fatality rates.

prediction <- function(iters, alpha, beta){
 out <- data.frame(collision = rep(NA,iters),
                   exposure = rep(NA, iters),
                   fatality = rep(NA, iters)
                   )
 for(n in 1:iters){
   c <- rbeta(1, shape1 = 9.38, shape2 = 3224.51)
   e <- rgamma(1, shape = alpha, rate = beta)
   f <- c*e
   out[n,] <- c(c,e,f)
 }
 return(out)
}

We generated predictions at each survey site using only data from that site, in which case a and b were set to the observed values, and by updating the prior exposure distribution with observed values following Bay et al. (2016). We applied this same procedure to all 1000 combinations of simulated survey effort and eagle exposure, multiplying the predicted fatality rates by the mean expansion factor derived from survey data to obtain distributions of predicted annual fatalities.

#Predictions integrating Exposure prior and survey data
prediction(100000, a + mean(Bay16$FLIGHT_MIN), b + mean(Bay16$EFFORT))

#Predictions from survey data
prediction(100000, a, b)